Getting a flat, thin, long, cube (ribbon) to follow Beizer curve?
I would make the curve flat by not bending a mesh to fit it, but "beveling" the curve (adding thickness).


Add a BezierCircle (in addition to the curve you already have)
In edit mode convert the handle type to Vector (super quick way V,V), that will make it a square.
Rotate it 45° along the Z axis (just to make it straight along an axis).
Scale the square along the X axis, to make it a thin ribbon.
(Last step) On your other curve object set the Bevel Object to be the "BezierCircle" you added in step one.